16.522: Session16
Session 18
Courtesy of Prof. Eduardo Ahedo
Universidad Politecnica de Madrid
HALL THRUSTERS:
FLUID MODEL
OF THE
DISCHARGE
Electric Space Propulsion, E. Ahedo
3.B.1
Types of models
•
•
•
•
Kinetic: based on Boltzmann eq.; unaffordable except for particular
aspects of the problem
Fluid: familiar formulation but important difficulties arising from
1. Weak collisionality (Kn large)
2. Wall interaction
3. Curved magnetic topology
4. 2D subsonic/supersonic ion flows
Particle In Cell & MonteCarlo methods (PIC/MCC): good for weak
collisionality; simple to implement, but subject to ‘numerical effects’;
important difficulties in dealing with disparate scales of electron and ions.
MCC model collisions statistically.
Hybrid (PIC/MCC for heavy species & fluid for electrons): best
compromise today; allows 2D (geometrical and magnetic) effects; avoids
small electron scales and admits quasineutrality
Electric Space Propulsion, E. Ahedo
3.B.2
Simulation of the plasma discharge
•
•
•
•
•
•
•
•
•
2D, axisymmetric model
Quasineutral plasma except for sheaths around walls.
Plasma wall interaction treated in separate sheath models.
Boundaries: 1) anode + gas injector, 2) cathode surface,
3) lateral walls
3 or 4 species: neutrals, ions (+, ++), and electrons
Ion dynamics: unmagnetized, near collisionless, internal
regular sonic transitions, singular sonic transitions at
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collisional (local thermodynamic equilibrium is not
assured)
Fluid modelling: complex and uncertain.
There is no fully 2D model yet. Two existing options:
a) Approximate 1D (axial) fluid model
b) Near 2D hybrid model: fluid eqs. for electrons;
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particle model for ions & neutrals
excluded from our Creative Commons license. For more information,
see http://ocw.mit.edu/help/faq-fair-use/.
Electric Space Propulsion, E. Ahedo
3.B.3
2D fluid model
∂
= 0, uθ i = 0, uθ n = 0
∂θ
⇒ boundary conditions at sheath edges
• Basic hypotheses: 1) azimuthal symmetry:
2) Quasineutrality: ne = ni + + 2ni ++
3) Simplified treatment of pressure tensors
• Continuity equations:
∂ne
+ ∇ ⋅ ne ue = Sion
∂t
∂ni
+ ∇ ⋅ ni ui = Sion ,
∂t
• Electron momentum:
∂nn
+ ∇ ⋅ nn un = − Sion
∂t
(here ne = ni only)
∂
( me ne ue ) + ∇ ⋅ me ne ue ue = −∇(neTe ) − ene ( E + ue × B ) + M e
∂t
M e ≈ − ne nn Ren (Te )me ue
• Ion momentum:
Sion ≈ ne nn Rion (Te ) ← ionization
←
e-n momentum transfer
∂
( mi ni ui ) + ∇ ⋅ mi ni ui ui = −∇(niTi ) + eni E + M i
∂t
M i ≈ Sion mi un − Scx mi (ui − un ) ←
ionization + charge-exchange mom. transf.
Electric Space Propulsion, E. Ahedo
3.B.4
2D fluid model
i Electron (total) energy equation :
∂ ⎛3
1
1
⎡⎛ 5
⎤
2⎞
2⎞
+
+
∇
⋅
+
+
T
m
u
n
T
m
u
n
u
q
e e ⎟ e
e e ⎟ e e
e ⎥ = − ene ue ⋅ E + Qe
⎜ e
⎜ e
⎢
∂t ⎝ 2
2
2
⎠
⎠
⎣⎝ 2
⎦
Qe ≈ −ne nn Rion (Te ) Eionα ion ← ionization
( Eion = 12.1eV, α ion ∼ 1.5 − 2.5 for Xe)
i Internal energy = total energy − mechanical energy
∂ ⎛3
⎞
⎡3
⎤
T
n
+
∇
⋅
T
n
u
+
q
= − neTe ∇ ⋅ ue + Q 'e
e
e
e
e
e
e
⎜
⎟
⎢
⎥
∂t ⎝ 2
⎠
⎣2
⎦
1
1
⎡⎛
⎤
⎞
Q 'e ≡ Qe − M e ⋅ ue + me ue 2 Se ≈ ne nn ⎢⎜ Ren + Rion ⎟ me ue 2 − Rion Eionα ion ⎥
2
2
⎠
⎣⎝
⎦
i Heat flux (diffusive model): (Bittencourt)
5
pe ∇Te − eqe ∧ B − meν e qe
2
i Energy equations for ions and neutrals.
o=−
→
qe = −Κ e ⋅ ∇Te
Electric Space Propulsion, E. Ahedo
3.B.5
From 2D to 1D model
∂ne ∂ne uze 1 ∂rne ure
+
+
= neν i
∂t
∂z
r ∂r
i The 1D axial model works with values averaged over each ('doughnut') radial section
i Electron continuity equation:
0=
2
rext2 − rint2
∫
rext
rint
⎤
⎡ ∂ne ∂ne uze 1 ∂rne ure
+
+
−
ν
n
e i⎥
∂z
r ∂r
⎦
⎣⎢ ∂t
→
∂ne ∂ne uze
+
= ne (ν i − ν w )
∂t
∂z
ext
( rne ure ) |int
with ne ≡ ne ( z ),..., and 'source' neν w = 2 2
evaluating losses at lateral walls.
rext − rint2
i An auxiliary radial model (at each z ) is needed to
1 ∂rne ure
≈ neν w
r ∂r
i This is a variable-separation type of solution;
ν w ( z ) is an eigenvalue of the radial model.
compute ν w & determine radial profiles:
i We proceed similarly with the rest of fluid equations.
Electric Space Propulsion, E. Ahedo
3.B.6
1D axial model
i
Neglect jet divergence, doubly-charged ions, pi , pn ,...
i Wall interaction terms appear as source terms (instead of BCs)
d
( ne uzi ) = ne (ν i − ν w )
dz
ν i : ionization frequency ν w : recombination frequency
i Conservation of species flows:
nn uzn + ne uzi = const = m A / Ami
ne uzi − ne uze = const = I d / Ae
i Ion axial momentum equation:
d
dφ
+ mi ne (ν i uzn − ν w uzi )
( mi ne uzi 2 ) = − ene
dz
dz
i Neutral axial momentum equation
d
( mi nn uzn 2 ) = mi ne (ν wn uznw − ν i uzn )
dz
uznw velocity of neutrals from recombination ( ≠ uzi )
Electric Space Propulsion, E. Ahedo
3.B.7
1D axial model
−uze =
i Electron Ohm´s law:
νe
meωe2
⎡ dφ 1 d
⎤
e
(
n
T
)
−
⎢ dz n dz e e ⎥
e
⎣
⎦
uθ e = −uze
ωe
νe
ν e = ν en + ν ei + ν wm + ν turb : effective collision frequency
(ν wm due to wall interaction, ν turb due to plasma turbulence)
i Electron internal energy:
d ( neTe )
d ⎛5
⎞
+
=
+ ν e ne me ue2 − ν i ne Eionα ion − β eν w neTe
n
T
u
q
u
ze ⎟
ze
⎜ e e ze
dz ⎝ 2
dz
⎠
( β e ∼ 6 − 100: factor for energy losses at lateral walls)
i Heat conduction: 0 = −
5
pe ∇Te − eqe ∧ B − meν e qe
2
i Normalized set of equations: (Te − mi uzi2 )
→
qze = −
5neTe ν e dTe
2me ωe2 dz
dY
= F (Y )
dz
Y = ( ne , nn , uzi , uze , uzn , Te , qze , φ ) vector of 8 plasma variables
i Singular (sonic) points of the equations at : M ≡
uzi
Te / mi
Electric Space Propulsion, E. Ahedo
= ±1
3.B.8
1D axial model
i For instance:
duzi
d ln A
G
= ν i − uzi
− uzi
dz
dz
Te − mi uzi2
⎛
2 qze ⎞
ωe2
2 d ln A
−
−
+
(2
)
G = − me uze ⎜ 1 −
m
u
u
m
u
ν
⎟ i i
zi
zn
i zi
5
n
T
u
dz
νe
e e ze ⎠
⎝
i A singular transition exists at anode sheath edge (point B): M B = −1 & GB = ∞
2
ω
υe
−e
i A regular subsonic/supersonic transition exists inside the channel (point S) with
M S = 1 & F (YS ) = 0, i.e. GS = 0
i Point S is equivalent to the critical section of a Laval nozzle, where
0=GS ≡ mi uzi2 d ln A / dz corresponds to dA / dz = 0. In a Hall thruster, GS =0
corresponds to a certain balance between ionization and electron diffusion.
i 8 boundary conditions are set at points B, S and P:
- Discharge voltage (1)
- Density and velocity of injected gas at anode (2)
- Electron temperature at cathode (1)
- Regularity condition at point S (1)
- Anode sheath conditions (3)
Electric Space Propulsion, E. Ahedo
3.B.9
Anode sheath in a Hall thruster
• For Maxwellian-type VDF electron flux to anode ∼ neA Te / 2π me ,
• In normal conditions this is much larger than
the quasineutral flux in the channel, g ze ≡ ne uze ( > 0)
• A negative anode sheath AB is formed in order to
satisfy g zeB =g zeA :
⎛ eφ ⎞ TeB
neB uzeB ≈ −neB exp ⎜ − AB ⎟
⎝ TeB ⎠ 2π me
This equation determines the sheath potential fall, φAB .
• Bohm condition at sheath edge: uziB ≈ − TeB / mi
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Commons license. For moreinformation,
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• Electron energy flux deposited: a) at the anode A,
1
tot
qzeA
= − ∫ me w2 wz f eB ( w)d 3 w ≈ 2TeB g zeB
2
b) at the sheath edge B:
tot
qzeB
= g zeB (2TeB + eφAB )
• These are 3 boundary conditions for the axial
quasineutral model
Electric Space Propulsion, E. Ahedo
3.B.10
Axial structure
•
•
•
Acceleration region:
– Presents most of the potential drop & ion
acceleration
– For electrons: Joule heating competes
with wall cooling
– Heat conduction smooths Te profile
– Plasma production = plasma recombination
– Plasma density decreases (due to ion
acceleration)
Ionization region:
– Electron cooling due to ionization
– Maximum of plasma density inside
Ion backstreaming region:
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– Electric force very small
see http://ocw.mit.edu/help/faq-fair-use/.
– Ion reverse flow is small
• Magnetic field is adjusted for each case:
– Pressure drop (towards anode) dominates
- solid lines: 110V, 110G
electron diffusive flow
- dashed lines: 600V, 330G
– Length depends on ionization rate (i.e. Te)
Electric Space Propulsion, E. Ahedo
3.B.11
Axial structure
Influence of anode mass flow
Influence of magnetic field shape
External B-field profiles
- Solid lines:
4 mg/s
- Dashed lines: 10 mg/s
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excluded from our Creative Commons license. For more information,
see http://ocw.mit.edu/help/faq-fair-use/.
© American Institute of Physics. All rights reserved. This content is
excluded from our Creative Commons license. For more information,
see http://ocw.mit.edu/help/faq-fair-use/.
Electric Space Propulsion, E. Ahedo
3.B.12
Thrust
• Plasma momentum flow: Fp ( z ) =
∑
α
=i ,e ,n
Fα ( z ), Fα ( z ) = ( mα uz2α + Tα ) nα A
• Axial momentum equation for the plasma:
dFi
dφ
= − Aeni
+ Ami ne (ν i uzn − ν w uzi )
dz
dz
Electric force
Ionization + wall recombination
dFn
= Ami ne (ν w uznw − ν i uzn )
dz
Magnetic force
dFe
dφ
dA
= Aene
+ Ajθ e Br + neTe
dz
dz
dz
⇒
dFp
dz
=
ε 0 d ⎛ dφ ⎞
Jet divergence
2
dA
(
)
+
−
−
+
ν
Aj
B
A
m
u
u
n
n
T
θe r
w i
zi
znw
e
e e
⎜
⎟
2 dz ⎝ dz ⎠
dz
Non-neutral
E-field force
Electric Space Propulsion, E. Ahedo
3.B.13
Thrust
• Integrating the preceding equation between anode and far downstream:
• Thrust:
F = Fp∞ − D plume = Fmag + FpA − Felec − Dwall
∞
− Magnetic force: Fmag = ∫ jθ e Br Adz
A
− Electric force: Felec
⎡ ε 0 ⎛ dφ ⎞
= ⎢A ⎜
⎟
2
dz
⎝
⎠
⎢⎣
2
⎤
⎥
⎥⎦ A
E
− Ion wall impact drag: Dwall = ∫ ν w mi (uzi − uznw )ne Adz
A
∞
dA
∫A dz dz
• Therefore, thrust is transmitted to the thruster through the magnetic
reaction force of electrons on the thruster magnetic circuit.
− Notice the contribution of the external magnetic field to thrust.
− Jet divergence drag: D plume =
neTe
− Pressure forces at the anode make a marginal contribution
− Ion energy accommodation at walls acts as a drag force.
Electric Space Propulsion, E. Ahedo
3.B.14
Bibliography
1. Ahedo, Presheath/sheath model of a plasma with secondary emission from two
parallel walls, Physics of Plasmas, vol. 9, pp. 4340--4347, 2002.
2. Ahedo, Gallardo, Martínez-Sánchez, Effects of the radial-plasma wall interaction
on the axial Hall thruster discharge, Physics of Plasmas, vol. 10, pp. 3397--3409,
2003.
3. Ahedo, Escobar, Influence of design and operation parameters on Hall thruster
performances, Journal of Applied Physics, vol. 96, pp. 983--992, 2004.
4. Barral, Makowski, Peradzynski, Gascon, Dudeck, Wall material effects in
stationary plasma thrusters. II. Near-wall and in-wall conductivity, Phys. Plasmas,
vol. 10, pp.4137 -- 4152, 2003.
5. Boeuf, Garrigues, Low frequency oscillations in a stationary plasma thruster, J.
Applied Physics, vol. 84, pp. 3541-3554, 1998.
Electric Space Propulsion, E. Ahedo
3.B.18
HALL THRUSTERS:
HYBRID 2D
CODE
Electric Space Propulsion, E. Ahedo
3.C.1
Particle-in-cell (PIC) methods
•
•
•
Individual motion of macroparticles, subject to electromagnetic forces, is
followed in a computational grid. (see Birdsall-Langdon)
Cell size, lcell, smaller than plasma gradients
Timestep, ∆τ, such that particles advance less than cell size.
•
Number of macroparticles per cell, Ncell, depends
on good statistics and small numerical oscillations.
•
Mass of macroparticle depends on species density
•
Different species Æ different sets of macroparticles
•
Example for Hall thruster (only ions and neutrals)
© IEPC-2005-040. All rights reserved. This content is
excludedfrom our Creative Commons license. For more
information, seehttp://ocw.mit.edu/help/faq-fair-use/.
– axisymmetric, 30mm × 15 mm Æ 900 (toroidal) cells Æ lcell~0.5mm
– Ncell (per species) ~ 30
Æ
27.000 macroparticles/species
Æ 109 -1011 atoms/macroparticle
– macroparticle mass ~ atom mass × (atom/macroparticle)
– ∆τ ~ 10-8 s (ions), 10-7 s (neutrals), 10-10 s (electrons)
Electric Space Propulsion, E. Ahedo
3.C.2
Video of particle motion
Courtesy of Professor Ahedo, Universidad Carlos III de Madrid. Used with permission.
Electric Space Propulsion, E. Ahedo
3.C.7
Video of plasma dynamics
Courtesy of Professor Ahedo, Universidad Carlos III de Madrid. Used with permission.
Time-averaged 2D behavior
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6LPXODWLRQRI+LJK6SHFLILF,PSXOVHDQG'RXEOH6WDJH+DOO7KUXVWHUV,Q3URFWK,QWHUQDWLRQDO(OHFWULF
3URSXOVLRQ&RQIHUHQFH3ULQFHWRQ86$
Electric Space Propulsion, E. Ahedo
3.C.9
Time-averaged 2D behavior
B streamlines
Te=const along
magnetic lines
Ion recombination
at walls & neutral
re-emission
&RXUWHV\RI3URIHVVRU$KHGR8QLYHUVLGDG&DUORV,,,GH0DGULG8VHGZLWKSHUPLVVLRQ
Electric Space Propulsion, E. Ahedo
3.C.10
Bibliography
1. Escobar, Antón, Ahedo. Simulation of high-specific-impulse and doublestage Hall thrusters, IEPC-2005-040, 2005.
2. Hagelaar, Bareilles, Garrigues, Boeuf, Two-dimensional model of a
stationary plasma thruster, Journal of Applied Physics, vol. 91, pp. 55925598, 2002.
3. Parra, Ahedo, Fife, Martínez-Sánchez, A two-dimensional hybrid model of
the Hall thruster discharge, Journal of Applied Physics,
Electric Space Propulsion, E. Ahedo
3.C.13
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16.522 Space Propulsion
Spring 2015
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